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Gordon Slade

Publications and source records attributed to Gordon Slade.

At least 55 records · Page 3Linked to original sources
arXiv2014

Structural stability of a dynamical system near a non-hyperbolic fixed point

We prove structural stability under perturbations for a class of discrete-time dynamical systems near a non-hyperbolic fixed point. We reformulate the stability problem in terms of the well-posedness of an infinite-dimensional nonlinear ordinary differential equation in a Banach space of carefully weighted sequences. Using this, we prove existence and regularity of flows of the dynamical system which obey mixed initial and final boundary conditions. The class of dynamical systems we study, and the boundary conditions we impose, arise in a renormalisation group analysis of the 4-dimensional weakly self-avoiding walk and the 4-dimensional n-component $|φ|^4$ spin model.

math.DS↗
arXiv2014

Scaling limits and critical behaviour of the 4-dimensional n-component $|φ|^4$ spin model

We consider the $n$-component $|φ|^4$ spin model on $\mathbb{Z}^4$, for all $n \geq 1$, with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent $\frac{n+2}{n+8}$ for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for $n =1,2,3$; double logarithmic scaling for $n=4$; and is bounded when $n>4$. In addition, for the model defined on the $4$-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the $|φ|^4$ model.

math-ph↗
arXiv2012

Lectures on Self-Avoiding Walks

These lecture notes provide a rapid introduction to a number of rigorous results on self-avoiding walks, with emphasis on the critical behaviour. Following an introductory overview of the central problems, an account is given of the Hammersley--Welsh bound on the number of self-avoiding walks and its consequences for the growth rates of bridges and self-avoiding polygons. A detailed proof that the connective constant on the hexagonal lattice equals $\sqrt{2+\sqrt{2}}$ is then provided. The lace expansion for self-avoiding walks is described, and its use in understanding the critical behaviour in dimensions $d>4$ is discussed. Functional integral representations of the self-avoiding walk model are discussed and developed, and their use in a renormalisation group analysis in dimension 4 is sketched. Problems and solutions from tutorials are included.

math.PR↗
arXiv2011

The strong interaction limit of continuous-time weakly self-avoiding walk

The strong interaction limit of the discrete-time weakly self-avoiding walk (or Domb--Joyce model) is trivially seen to be the usual strictly self-avoiding walk. For the continuous-time weakly self-avoiding walk, the situation is more delicate, and is clarified in this paper. The strong interaction limit in the continuous-time setting depends on how the fugacity is scaled, and in one extreme leads to the strictly self-avoiding walk, in another to simple random walk. These two extremes are interpolated by a new model of a self-repelling walk that we call the "quick step" model. We study the limit both for walks taking a fixed number of steps, and for the two-point function.

math.PR↗
arXiv2011

The growth constants of lattice trees and lattice animals in high dimensions

We prove that the growth constants for nearest-neighbour lattice trees and lattice (bond) animals on the integer lattice Zd are asymptotic to 2de as the dimension goes to infinity, and that their critical one-point functions converge to e. Similar results are obtained in dimensions d>8 in the limit of increasingly spread-out models; in this case the result for the growth constant is a special case of previous results of M. Penrose. The proof is elementary, once we apply previous results of T. Hara and G. Slade obtained using the lace expansion.

math.PR↗
arXiv2009

Functional integral representations for self-avoiding walk

We give a survey and unified treatment of functional integral representations for both simple random walk and some self-avoiding walk models, including models with strict self-avoidance, with weak self-avoidance, and a model of walks and loops. Our representation for the strictly self-avoiding walk is new. The representations have recently been used as the point of departure for rigorous renormalization group analyses of self-avoiding walk models in dimension 4. For the models without loops, the integral representations involve fermions, and we also provide an introduction to fermionic integrals. The fermionic integrals are in terms of anti-commuting Grassmann variables, which can be conveniently interpreted as differential forms.

math.PR↗
arXiv2008

Invasion percolation on regular trees

We consider invasion percolation on a rooted regular tree. For the infinite cluster invaded from the root, we identify the scaling behavior of its $r$-point function for any $r\geq2$ and of its volume both at a given height and below a given height. We find that while the power laws of the scaling are the same as for the incipient infinite cluster for ordinary percolation, the scaling functions differ. Thus, somewhat surprisingly, the two clusters behave differently; in fact, we prove that their laws are mutually singular. In addition, we derive scaling estimates for simple random walk on the cluster starting from the root. We show that the invasion percolation cluster is stochastically dominated by the incipient infinite cluster. Far above the root, the two clusters have the same law locally, but not globally. A key ingredient in the proofs is an analysis of the forward maximal weights along the backbone of the invasion percolation cluster. These weights decay toward the critical value for ordinary percolation, but only slowly, and this slow decay causes the scaling behavior to differ from that of the incipient infinite cluster.

math.PR↗
arXiv2007

Random walk on the incipient infinite cluster for oriented percolation in high dimensions

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on $Z^d \times Z_+$. In dimensions $d>6$, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4/3, and thereby prove a version of the Alexander--Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions $d>6$, by extending results about critical oriented percolation obtained previously via the lace expansion.

math.PR↗
arXiv2007

An extension of the inductive approach to the lace expansion

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6.

math.PR↗
arXiv2007

Extension of the generalised inductive approach to the lace expansion: Full proof

This paper extends the inductive approach to the lace expansion of van der Hofstad and Slade in order to prove Gaussian asymptotic behaviour for models with critical dimension other than 4. The results are applied by Holmes to study sufficiently spread-out lattice trees in dimensions d>8 and may also be applicable to percolation in dimensions d>6.

math.PR↗
arXiv2004

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $Ω$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,λ)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λV^{1/3}$, where $λ$ is fixed and positive. We show that for any such model, there is a phase transition at $p_c$ analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold $p_c$. In particular, we show that the largest cluster inside a scaling window of size $|p-p_c|=Θ(\cn^{-1}V^{-1/3})$ is of size $Θ(V^{2/3})$, while below this scaling window, it is much smaller, of order $O(ε^{-2}\log(Vε^3))$, with $ε=\cn(p_c-p)$. We also obtain an upper bound $O(\cn(p-p_c)V)$ for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order $Θ(\cn(p-p_c))$. Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the $n$-cube and certain Hamming cubes, as well as the spread-out $n$-dimensional torus for $n>6$.

math.PR↗
arXiv2004

Random subgraphs of finite graphs: II. The lace expansion and the triangle condition

In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set $\{0,1,..., r-1\}^n$, including the Hamming cube on an alphabet of $r$ letters (the $n$-cube, for $r=2$), the $n$-dimensional torus with nearest-neighbor bonds and $n$ sufficiently large, and the $n$-dimensional torus with $n>6$ and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

math.PR↗
arXiv2004

Random subgraphs of finite graphs: III. The phase transition for the $n$-cube

We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λ2^{n/3}$, where $λ$ is a small positive constant. Let $ε=n(p-p_c(n))$. In two previous papers, we showed that the largest cluster inside a scaling window given by $|ε|=Θ(2^{-n/3})$ is of size $Θ(2^{2n/3})$, below this scaling window it is at most $2(\log2) nε^{-2}$, and above this scaling window it is at most $O(ε2^n)$. In this paper, we prove that for $p - p_c(n) \geq e^{-cn^{1/3}}$ the size of the largest cluster is at least $Θ(ε2^n)$, which is of the same order as the upper bound. This provides an understanding of the phase transition that goes far beyond that obtained by previous authors. The proof is based on a method that has come to be known as ``sprinkling,'' and relies heavily on the specific geometry of the $n$-cube.

math.PR↗
arXiv2004

Expansion in $n^{-1}$ for percolation critical values on the $n$-cube and $Z^n$: the first three terms

Let $p_c(\mathbb{Q}_n)$ and $p_c(\mathbb{Z}^n)$ denote the critical values for nearest-neighbour bond percolation on the $n$-cube $\mathbb{Q}_n = \{0,1\}^n$ and on $\Z^n$, respectively. Let $Ω= n$ for $\mathbb{G} = \mathbb{Q}_n$ and $Ω= 2n$ for $\mathbb{G} = \mathbb{Z}^n$ denote the degree of $\mathbb{G}$. We use the lace expansion to prove that for both $\mathbb{G} = \mathbb{Q}_n$ and $\mathbb{G} = \mathbb{Z}^n$, $p_c(\mathbb{G}) & = \cn^{-1} + \cn^{-2} + {7/2} \cn^{-3} + O(\cn^{-4}).$ This extends by two terms the result $p_c(\mathbb{Q}_n) = \cn^{-1} + O(\cn^{-2})$ of Borgs, Chayes, van der Hofstad, Slade and Spencer, and provides a simplified proof of a previous result of Hara and Slade for $\mathbb{Z}^n$.

math.PR↗
arXiv2000

A generalised inductive approach to the lace expansion

The lace expansion is a powerful tool for analysing the critical behaviour of self-avoiding walks and percolation. It gives rise to a recursion relation which we abstract and study using an adaptation of the inductive method introduced by den Hollander and the authors. We give conditions under which the solution to the recursion relation behaves as a Gaussian, both in Fourier space and in terms of a local central limit theorem. These conditions are shown elsewhere to hold for sufficiently spread-out models of networks of self-avoiding walks in dimensions $d>4$, and for critical oriented percolation in dimensions $d+1>5$, providing a unified approach and an essential ingredient for a detailed analysis of the branching behaviour of these models.

math.PR↗